Spectral convergence of Riemannian manifolds. II
type: Item
instance of: scholarly article
title: Spectral convergence of Riemannian manifolds. II
zbMATH Open document ID: 0853.58100
author: Atsushi Kasue
author: Hironori Kumura
published in: Tôhoku Mathematical Journal. Second Series
publication date: 1996-07-18T00:00:00Z
review text: Let \({\mathcal M}_c\) denote the isometry classes of compact connected Riemannian manifolds \(M\). Let \(\rho_M (t,x,y)\) be the heat kernel of the Laplace operator of \(M\) relative to the normalized measure \(\text{dvol}_M/\text{vol}(M)\). The spectral distance \(\text{SD}(M,N)\) measures the difference between \(\rho_M(t,x,y)\) and \(\rho_N(t,f(x),f(y))\) where the infimum is taken over all maps \(f: M \to N\) and is suitably normalized in \(t\); it gives a uniform structure to \({\mathcal M}_c\). The Hausdorff distance \(\text{HD} (M,N)\) is another uniform structure which was introduced by Gromov. The authors prove precompactness theorems concerning both SD and HD and they study the completion of a SD precompact family. [For part I see the authors, ibid. 46, No. 2, 147-179 (1994; Zbl 0814.53035)].
reviewed by: Peter B. Gilkey
Mathematics Subject Classification ID: 58J35
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 889576
zbMATH Keywords: spectral distance
zbMATH Keywords: Hausdorff distance
zbMATH Keywords: heat kernel
zbMATH Keywords: compactness theorems
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2748/tmj/1178225413
OpenAlex ID: W2023222785
Wikidata QID: Q115227262
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DOI: 10.2748/TMJ/1178225413
description: scientific article; zbMATH DE number 889576